Actions

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Taking into account all the previous suggestions, the assumed values are:

- c _, = 18 mm for the beams and c _, = 20 mm for the columns, according to the design bars’ diameter (i.e. 𝜙18 for the beams and 𝜙20 for the columns);

- c _, = 25 𝑚𝑚, being the structural class S4 and the exposure coefficient XC2;

- ∆𝑐 _, = ∆𝑐 _, = ∆𝑐 _, = 0 𝑚𝑚

Thus, according to (3.3), the minimum concrete cover is 𝑐 = max{20; 25; 10} = 25mm.

In conclusion, considering that the recommended value of ∆𝑐 is 10𝑚𝑚

according to §4.4.1.2 of Eurocode 2 [28], referring to equation (3.2), 𝑐 = 𝑐 + ∆𝑐 = 25 + 10 = 35 𝑚𝑚, used both for columns and beams.

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- Fundamental combination, usually assumed for Ultimate Limit State (ULS):

𝛾 𝐺 + 𝛾 𝐺 + 𝛾 𝑃 + 𝛾 𝑄 + 𝛾 Ψ 𝑄 + 𝛾 Ψ 𝑄 +... (3.4) - Rare combination, usually assumed for irreversible Serviceability Limit

State (SLS):

𝐺 + 𝐺 + 𝑃 + 𝑄 + Ψ 𝑄 + Ψ 𝑄 +.. (3.5)

- Frequent combination, usually assumed for reversable Serviceability Limit State (SLS):

𝐺 + 𝐺 + 𝑃 + Ψ 𝑄 + Ψ 𝑄 + Ψ 𝑄 +.. (3.6)

- Quasi-permanent combination (SLS), usually assumed for reversable long term effects:

𝐺 + 𝐺 + 𝑃 + Ψ 𝑄 + Ψ 𝑄 + Ψ 𝑄 +.. (3.7)

- Accidental combination, usually assumed for Serviceability Limit State (SLS) and long-term effects:

𝐺 + 𝐺 + 𝑃 + A + Ψ 𝑄 + Ψ 𝑄 + Ψ 𝑄 +.. (3.8) - Seismic combination, usually assumed for both SLS and ULS, when

considering seismic actions:

𝐺 + 𝐺 + 𝑃 + Ψ 𝑄 + Ψ 𝑄 +.. (3.9)

where 𝛾 , 𝛾 and 𝛾 are the partial coefficient of respectively permanent structural loads, permanent non-structural loads and live loads, while Ψ are the combination coefficients related to the j^th variable action. The latter depend on the type of action, on the category of the structure and on the design situation, as defined in table A1.1 of Eurocode 0 [5], here reported in Table 3.10:

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Table 3.10: Combination coefficients

3.5.1 Permanent actions

3.5.1.1 Permanent structural loads (𝑮_𝟏)

The permanent structural loads (𝐺 ) are defined by the self-weight of the beams and the columns, and the self-weight of the reinforced concrete and hollow tiles mixed floor slab. The former are directly computed by the FEM software, function of the specific weight (25 𝑘𝑁/𝑚 for the reinforced concrete and 24 𝑘𝑁/𝑚 for the concrete only) and the cross-section of the beams and the columns, while the latter is computed taking into account the slab scheme (Figure 3.3).

Figure 3.3: Slab scheme (dimensions in cm)

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Considering all the contributions (Table 3.11), the permanent structural load of the slab is equal to 3.20 𝑘𝑁/𝑚 .

Table 3.11: Permanent structural load of the slab

𝑾𝒊𝒅𝒕𝒉 [𝒎] 𝑻𝒉𝒊𝒄𝒌𝒏𝒆𝒔𝒔[𝒎] 𝑾𝒆𝒊𝒈𝒉𝒕

[𝒌𝑵/𝒎^𝟑] 𝒈_𝟏 [𝒌𝑵/𝒎^𝟐]

Slab 1.00 0.05 25 1.25

Rib 2x0.10 0.18 25 0.90

Brick 2x0.40 0.18 7.3 1.05

3.20

3.5.1.2 Permanent non-structural load (𝑮_𝟐)

The permanent non-structural load (𝐺 ) depends on the non-structural parts of the slabs (screed, floor and plaster) and the inner walls. The former is equal to 1.40 𝑘𝑁/𝑚 , as Table 3.12 shows:

Table 3.12: Permanent non-structural load of the slab

𝑾𝒊𝒅𝒕𝒉 [𝒎] 𝑻𝒉𝒊𝒄𝒌𝒏𝒆𝒔𝒔[𝒎] 𝑾𝒆𝒊𝒈𝒉𝒕

[𝒌𝑵/𝒎^𝟑] 𝒈_𝟐 [𝒌𝑵/𝒎^𝟐]

Screed 1.00 0.05 16.0 0.8

Floor - - - 0.20

Plaster 1.00 0.02 20.0 0.40

1.40

The other part of 𝐺 , related to the internal walls, can be computed considering the Figure 3.4:

Figure 3.4: Scheme of the internal walls (dimensions in cm)

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Considering the specific weight of the elements composing the internal walls, it is possible to obtain the permanent non-structural load as follows (Table 3.13):

Table 3.13: Internal walls’ weight

𝑾𝒊𝒅𝒕𝒉 [𝒎] 𝑾𝒆𝒊𝒈𝒉𝒕 [𝒌𝑵/𝒎^𝟑] 𝒈_𝟐 [𝒌𝑵/𝒎^𝟐]

Brick 0.08 6.0 0.48

Plaster 0.01 20.0 0.40

0.88

To calculate the 𝐺 value associated to the internal walls, code rules suggest loads per square meters to refer, starting from the load per meter. This value can be easily computed for the specific case by multiplying the previous load for the height of the floor, as follows:

𝐺 = 2.65 ∙ 0.88 = 2.33𝑘𝑁/𝑚 (3.10)

Thus, considering the DM 2018 [9] at § 3.1.3, the 𝑔 value can be assumed equal to 1.20 𝑘𝑁/𝑚 :

Table 3.14: Permanent non-structural load for internal walls

The total permanent non-structural load is then equal to 𝒈_𝟐 = 𝟐. 𝟔𝟎 𝒌𝑵/𝒎^𝟐.

3.5.2 Variable actions

3.5.2.1 Live loads

This is the overload due to the use of the building, and, referring to table 3.1.II of DM2018 [9], it is equal to 2.00 𝑘𝑁/𝑚 for the floors and 0.50 𝑘𝑁/𝑚 for the roofing.

3.5.2.2 Wind load

This action is defined in § 3.3 of DM2018 [9] as follows:

𝑝 = 𝑞 ∙ 𝑐 ∙ 𝑐 ∙ 𝑐 (3.11)

71 where:

- 𝑞 is kinetic wind pressure

The kinetic wind pressure 𝑞 is computed using the following formula:

𝑞 =1

2𝜌𝑣 (3.12)

Considering an air density 𝜌 = 1,25 𝑘𝑔/𝑚 and a reference wind velocity 𝑣 = 31.3 𝑚/𝑠 for L’Aquila city. Thus, by plug in these values on (3.12), it results 𝑞 = 612,3 𝑁/𝑚 .

- 𝑐 is the exposure factor

The exposure coefficient 𝑐 is given by the following formulae:

𝑐 (𝑧) = 𝑘 𝑐 ln(𝑧 𝑧⁄ ) [7 + 𝑐 ln(𝑧 𝑧⁄ )] for 𝑧 ≥ 𝑧 (3.13) 𝑐 (𝑧) = 𝑐 (𝑧 ) for 𝑧 < 𝑧 (3.14) where:

 𝑘 , 𝑧 , 𝑧 are given in table 3.3.II of DM2018 [9] depending on the exposure category of the site. Since the category is V, it results: 𝑘 = 0.23, 𝑧 = 0,70𝑚 , 𝑧 = 12𝑚

 𝑐 is the topography coefficient and the normative suggests 𝑐 = 1

Thus, it is possible to define the exposure coefficient as function of the height 𝑧, expressed in meters, by substituting the coefficients in (3.12) and (3.13):

Table 3.15: Exposure coefficient 𝑐 , as function of the height

𝒛 [𝒎] 0 3 6 9 12 15

𝒄_𝒆 [−] 1.48 1.48 1.48 1.48 1.48 1.63

- 𝑐 is the shape coefficient (or aerodynamic) and it is assumed equal to 0.80 for the upwind surface and −0.45 for the downwind surface

- 𝑐 is the dynamic factor, the normative suggests to assume 𝑐 = 1

In conclusion, by plug in all these coefficients in (3.11) and taking into account an influence area of 5 meters, it is possible to obtain the distribution of the wind pressure along the height:

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Table 3.16: Wind pressure, as function of the height

𝒛 [𝒎]

𝒑_𝒖𝒑 [𝒌𝑵/𝒎]

𝒑_{𝒅𝒐𝒘𝒏} [𝒌𝑵/𝒎]

0 3.60 -1.80

3 3.60 -1.80

6 3.60 -1.80

9 3.60 -1.80

12 3.60 -1.80

15 4.00 -2.00

3.5.2.3 Snow load

This action is defined in § 3.4 of DM2018 [9] as follows:

𝑞 = 𝜇 ∗ 𝑞 ∗ 𝐶 ∗ 𝐶 (3.15)

where:

- 𝑞 is the characteristic ground snow load and it depends on the location of the city (L’Aquila is in zone III) and on the elevation of the site (714 meters over the sea level for L’Aquila). Thus, referring to § 3.4.2, it results 𝑞 = 2.72𝑘𝑁/𝑚

- 𝜇 is the shape coefficient and depends on the inclination angle of the roofing. For the specific case, the roof is planar, thus referring to table 3.4.II of DM2018 [9], 𝜇 = 0.8

- 𝐶 is the exposure coefficient, suggested unitary in § 3.4.4 - 𝐶 is the thermal coefficient, suggested unitary in § 3.4.5 In conclusion, the snow load is 𝑞 = 2.17 𝑘𝑁/𝑚 for the case study.

3.5.3 Seismic action

This action is described in § 3.2 of DM2018 [9], where the design seismic action is defined as function of the seismic hazard of the site, associated to the morphological and stratigraphic characteristics of the ground where the structure is located.

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Seismic hazard is correlated to 𝑆𝑒(𝑇), that is the elastic horizontal ground acceleration response spectrum also called "elastic response spectrum”. This spectrum corresponds to a ground acceleration equal to the design ground acceleration on ground type A multiplied by the soil factor 𝑆, defined as:

𝑆 (𝑇) = 𝑓(𝑎 , 𝐹 , 𝑇^∗) (3.16)

where:

- 𝑎 is the design ground acceleration on ground type A - 𝐹 is the maximum horizontal amplification factor

- 𝑇^∗ is the corner period at the upper limit of the constant acceleration region of the elastic spectrum

To determine these values, the DM2018 [9] recommends the use of an Excel sheet, made available by the Ministry of Infrastructure and Transport. The procedure is divided into 3 phases.

The first phase (“fase 1” of the Excel Macro) regards the identification of the dangerousness of the site. Thus, it is needed to identify the region and city in which the building is placed. Then the geographical coordinates are automatically obtained.

The second phase (“fase 2” of the Macro) regards the choice of the design strategy. The input parameters are:

- Design working life (in year). The building is class 2 (buildings with ordinary level of performance) so 𝑉 = 50 𝑦𝑟

- Use coefficient 𝑐 = 1,0

In the last phase (“fase 3” of the Macro), the information are:

- Limit state considered: both ULS and SLS are considered;

- Ground type: 𝐵 “deposits of very dense sand, gravel, or very stiff clay, at least several tens of m in thickness, characterised by a gradual increase of mechanical properties with depth” (Table 3.1 of EC8 [29])

- Topographic class: 𝑇

- Behaviour factor 𝑞 = 4.5 ∙ 𝛼 /𝛼 = 4.5 ∙ 1.3 = 5.85, having chosen a ductility class A and 𝛼 /𝛼 = 1.3 for multi-storey frames

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- Structural factor 𝑞 = 𝐾 ∙ 𝑞 = 1 ∙ 5.85 = 5.85, having chosen a reduction factor 𝐾 = 1 as the normative suggests for regular structures in elevation - Damping factor: 𝜉 = 5% as conventionally assumed for reinforced concrete

structures

In the following, the response spectrum at ultimate limit state (Figure 3.5) and at the serviceability limit state (Figure 3.6) are represented.

Figure 3.5: Response spectrum at ULS

Figure 3.6: Response spectrum at SLS